zbMATH — the first resource for mathematics

A wavelet lifting approach to long-memory estimation. (English) Zbl 1384.62289
Summary: Reliable estimation of long-range dependence parameters is vital in time series. For example, in environmental and climate science such estimation is often key to understanding climate dynamics, variability and often prediction. The challenge of data collection in such disciplines means that, in practice, the sampling pattern is either irregular or blighted by missing observations. Unfortunately, virtually all existing H. E. Hurst [“Long-term storage capacity of reservoirs”, Trans. Am. Soc. Civil Eng. 116, 770–808 (1951)] parameter estimation methods assume regularly sampled time series and require modification to cope with irregularity or missing data. However, such interventions come at the price of inducing higher estimator bias and variation, often worryingly ignored. This article proposes a new Hurst exponent estimation method which naturally copes with data sampling irregularity. The new method is based on a multiscale lifting transform exploiting its ability to produce wavelet-like coefficients on irregular data and, simultaneously, to effect a necessary powerful decorrelation of those coefficients. Simulations show that our method is accurate and effective, performing well against competitors even in regular data settings. Armed with this evidence our method sheds new light on long-memory intensity results in environmental and climate science applications, sometimes suggesting that different scientific conclusions may need to be drawn.

62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62M15 Inference from stochastic processes and spectral analysis
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
Adlift; fArma; LASS; NESToolbox; nlt; R
Full Text: DOI
[1] Abry, P; Goncalves, P; Flandrin, P; Antoniadis, A (ed.); Oppenheim, G (ed.), Wavelets, spectrum analysis and \(1/f\) processes, No. 103, 15-29, (1995), New York · Zbl 0828.62083
[2] Abry, P; Flandrin, P; Taqqu, MS; Veitch, D; Park, K (ed.); Willinger, W (ed.), Wavelets for the analysis, estimation and synthesis of scaling data, 39-88, (2000), Chichester
[3] Abry, P., Goncalves, P., Véhel, J.L.: Scaling, Fractals and Wavelets. Wiley, New York (2013) · Zbl 1192.94005
[4] Beran, J., Feng, Y., Ghosh, S., Kulik, R.: Long-Memory Processes. Springer, New York (2013) · Zbl 1282.62187
[5] Bhattacharya, RN; Gupta, VK; Waymire, E, The Hurst effect under trends, J. Appl. Probab., 20, 649-662, (1983) · Zbl 0526.60027
[6] Blender, R; Fraedrich, K; Hunt, B, Millennial climate variability: GCM-simulation and greenland ice cores, Geophys. Res. Lett., 33, l04710, (2006)
[7] Broersen, P. M.T., De Waele, S., Bos, R. The accuracy of time series analysis for laser-doppler velocimetry, In: Proceedings of the 10th International Symposium Application of Laser Techniques to Fluid Mechanics (2000) · Zbl 0934.62094
[8] Broersen, P.M.T.: Time series models for spectral analysis of irregular data far beyond the mean data rate. Meas. Sci. Technol. 19, 1-13 (2007) · Zbl 0864.62061
[9] Claypoole, R.L., Baraniuk, R.G., Nowak, R.D.: Adaptive wavelet transforms via lifting. In: IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP), pp. 1513-1516. Seattle (1998)
[10] Clegg, RG, A practical guide to measuring the Hurst parameter, Int. J. Simul. Syst. Sci. Technol., 7, 3-14, (2006)
[11] Coeurjolly, J-F; Lee, K; Vidakovic, B, Variance estimation for fractional Brownian motions with fixed Hurst parameters, Commun. Stat. Theory Methods, 43, 1845-1858, (2014) · Zbl 1291.62056
[12] Coifman, RR; Donoho, DL; Antoniadis, A (ed.); Oppenheim, G (ed.), Translation-invariant de-noising, No. 103, 125-150, (1995), New York · Zbl 0866.94008
[13] Craigmile, PF; Percival, DB, Asymptotic decorrelation of between-scale wavelet coefficients, IEEE Trans. Image Process., 51, 1039-1048, (2005) · Zbl 1297.62191
[14] Craigmile, PF; Percival, DB; Guttorp, P; Casacuberta, C (ed.); Miró-Roig, RM (ed.); Verdera, J (ed.); Xambó-Descamps, S (ed.), The impact of wavelet coefficient correlations on fractionally differenced process estimation, 591-599, (2001), Basel · Zbl 1031.62071
[15] Dahlhaus, R, Efficient parameter estimation for self-similar processes, Ann. Stat., 17, 1749-1766, (1989) · Zbl 0703.62091
[16] Faÿ, G; Moulines, E; Roueff, F; Taqqu, MS, Estimators of long-memory: Fourier versus wavelets, J. Econom., 151, 159-177, (2009) · Zbl 1431.62367
[17] Flandrin, P, Wavelet analysis and synthesis of fractional Brownian motion, IEEE Trans. Image Process., 38, 910-917, (1992) · Zbl 0743.60078
[18] Flandrin, P.: Time-Frequency/Time-Scale Analysis. Academic Press, San Diego (1998) · Zbl 0926.94005
[19] Foster, G, Wavelets for period analysis of unevenly sampled time series, Astron. J., 112, 1709-1729, (1996)
[20] Fox, R; Taqqu, MS, Large-sample properties of parameter estimates for strongly dependent stationary Gaussian time series, Ann. Stat., 14, 517-532, (1986) · Zbl 0606.62096
[21] Fraedrich, K; Blender, R, Scaling of atmosphere and Ocean temperature correlations in observations and climate models, Phys. Rev. Lett., 90, 108501, (2003)
[22] Giraitis, L; Robinson, PM; Surgailis, D, Variance-type estimation of long memory, Stoch. Process. Appl., 80, 1-24, (1999) · Zbl 0955.62090
[23] Gloter, A; Hoffmann, M, Estimation of the Hurst parameter from discrete noisy data, Ann. Stat., 35, 1947-1974, (2007) · Zbl 1126.62073
[24] Granger, CWJ; Joyeux, R, An introduction to long-memory time series models and fractional differencing, J. Time Ser. Anal., 1, 15-29, (1980) · Zbl 0503.62079
[25] Grootes, PM; Stulver, M; White, JWC; Johnson, S; Jouzel, J, Comparison of oxygen isotope records from the GISP2 and GRIP greenland ice cores, Nature, 366, 552-554, (1993)
[26] Higuchi, T, Relationship between the fractal dimension and the power law index for a time series: a numerical investigation, Physica D, 46, 254-264, (1990) · Zbl 0719.62105
[27] Hsu, N-J, Long-memory wavelet models, Stat. Sin., 16, 1255-1271, (2006) · Zbl 1109.62077
[28] Hurst, HE, Long-term storage capacity of reservoirs, Trans. Am. Soc. Civil Eng., 116, 770-808, (1951)
[29] Jansen, M., Oonincx, P.: Second Generation Wavelets and Applications. Springer, Berlin (2005)
[30] Jansen, M; Nason, GP; Silverman, BW; Unser, M (ed.); Aldroubi, A (ed.), Scattered data smoothing by empirical Bayesian shrinkage of second generation wavelet coefficients, No. 4478, 87-97, (2001), Washington. DC
[31] Jansen, M; Nason, GP; Silverman, BW, Multiscale methods for data on graphs and irregular multidimensional situations, J. R. Stat. Soc. B, 71, 97-125, (2009) · Zbl 1231.62054
[32] Jensen, MJ, Using wavelets to obtain a consistent ordinary least squares estimator of the long-memory parameter, J. Forecast., 18, 17-32, (1999)
[33] Jeon, S; Nicolis, O; Vidakovic, B, Mammogram diagnostics via 2-D complex wavelet-based self-similarity measures, São Paulo J. Math. Sci., 8, 265-284, (2014) · Zbl 1369.92061
[34] Jung, YY; Park, Y; Jones, DP; Ziegler, TR; Vidakovic, B, Self-similarity in NMR spectra: an application in assessing the level of cysteine, J. Data Sci., 8, 1, (2010)
[35] Junger, WL; Ponce de Leon, A, Imputation of missing data in time series for air pollutants, Atmos. Environ., 102, 96-104, (2015)
[36] Kirchner, JW; Neal, C, Universal fractal scaling in stream chemistry and its implications for solute transport and water quality trend detection, Proc. Nat. Acad. Sci., 110, 12213-12218, (2013)
[37] Kiss, P; Müller, R; Jánosi, IM, Long-range correlations of extrapolar total ozone are determined by the global atmospheric circulation, Nonlinear Process. Geophys., 14, 435-442, (2007)
[38] Knight, MI; Nason, GP, A nondecimated lifting transform, Stat. Comput., 19, 1-16, (2009)
[39] Knight, M.I., Nunes, M.A.: nlt: a nondecimated lifting scheme algorithm, r package version 2.1-3 (2012) · Zbl 0934.62094
[40] Knight, MI; Nunes, MA; Nason, GP, Spectral estimation for locally stationary time series with missing observations, Stat. Comput., 22, 877-8951, (2012) · Zbl 1252.60034
[41] Lobato, I; Robinson, PM, Averaged periodogram estimation of long memory, J. Econom., 73, 303-324, (1996) · Zbl 0854.62088
[42] Lomb, N, Least-squares frequency analysis of unequally spaced data, Astrophys. Space Sci., 39, 447-462, (1976)
[43] Mandelbrot, BB; Taqqu, MS, Robust R/S analysis of long-run serial correlation, Bull. Int. Stat. Inst., 48, 59-104, (1979) · Zbl 0518.62036
[44] Mandelbrot, BB; Ness, JW, Fractional Brownian motions, fractional noises and applications, SIAM Rev., 10, 422-437, (1968) · Zbl 0179.47801
[45] Marvasti, F.A.: Nonuniform Sampling: Theory and Practice. Springer, New York (2001) · Zbl 0987.00028
[46] McCoy, EJ; Walden, AT, Wavelet analysis and synthesis of stationary long-memory processes, J. Comput. Graph. Stat., 5, 26-56, (1996)
[47] Meese, DA; Gow, AJ; Grootes, P; Stuiver, M; Mayewski, PA; Zielinski, GA; Ram, M; Taylor, KC; Waddington, ED, The accumulation record from the GISP2 core as an indicator of climate change throughout the holocene, Science, 266, 1680-1682, (1994)
[48] Nason, G; Silverman, B; Antoniadis, A (ed.); Oppenheim, G (ed.), The stationary wavelet transform and some statistical applications, No. 103, 281-300, (1995), New York · Zbl 0828.62038
[49] Nilsen, T; Rypdal, K; Fredriksen, H-B, Are there multiple scaling regimes in holocene temperature records?, Earth Syst. Dyn., 7, 419-439, (2016)
[50] Nunes, M.A., Knight, M.I.: Adlift: an adaptive lifting scheme algorithm. R package version 1.3-2 (2012) https://CRAN.R-project.org/package=adlift · Zbl 1102.35300
[51] Nunes, MA; Knight, MI; Nason, GP, Adaptive lifting for nonparametric regression, Stat. Comput., 16, 143-159, (2006)
[52] Palma, W.: Long-memory Time Series: Theory and Methods. Wiley, Chichester (2007) · Zbl 1183.62153
[53] Pelletier, JD; Turcotte, DL, Long-range persistence in climatological and hydrological time series: analysis, modeling and application to drought hazard assessment, J. Hydrol., 203, 198-208, (1997)
[54] Peng, C-K; Buldyrev, SV; Havlin, S; Simons, M; Stanley, HE; Goldberger, AL, Mosaic organization of DNA nucleotides, Phys. Rev. E, 49, 1685, (1994)
[55] Percival, DB; Guttorp, P, Long-memory processes, the allan variance and wavelets, Wavelets Geophys., 4, 325-344, (1994)
[56] Petit, J-R; Jouzel, J; Raynaud, D; Barkov, NI; Barnola, J-M; Basile, I; Bender, M; Chappellaz, J; Davis, M; Delaygue, G; etal., Climate and atmospheric history of the past 420,000 years from the vostok ice core, antarctica, Nature, 399, 429-436, (1999)
[57] R Core Team: R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna (2013) · Zbl 1294.62217
[58] Ramírez-Cobo, P; Lee, KS; Molini, A; Porporato, A; Katul, G; Vidakovic, B, A wavelet-based spectral method for extracting self-similarity measures in time-varying two-dimensional rainfall maps, J. Time Ser. Anal., 32, 351-363, (2011) · Zbl 1294.62217
[59] Rehfeld, K; Marwan, N; Heitzig, J; Kurths, J, Comparison of correlation analysis techniques for irregularly sampled time series, Nonlinear Process. Geophys., 18, 389-404, (2011)
[60] Rehman, S; Siddiqi, AH, Wavelet based Hurst exponent and fractal dimensional analysis of saudi climatic dynamics, Chaos Solitons Fractals, 40, 1081-1090, (2009)
[61] Rogozhina, I; Martinec, Z; Hagedoorn, JM; Thomas, M; Fleming, K, On the long-term memory of the greenland ice sheet, J. Geophys. Res., 116, f1, (2011)
[62] Scargle, J, Studies in astronomical time series analysis II-statistical aspects of spectral analysis of unevenly spaced data, Astrophys. J., 263, 835-853, (1982)
[63] Shi, B; Vidakovic, B; Katul, GG; Albertson, JD, Assessing the effects of atmospheric stability on the fine structure of surface layer turbulence using local and global multiscale approaches, Phys. Fluids, 17, 055104, (2005) · Zbl 1187.76481
[64] Shibata, Y; Shimizu, S, A decay property of the Fourier transform and its application to the Stokes problem, J. Math. Fluid Mech., 3, 213-230, (2001) · Zbl 1102.35300
[65] Stoev, S., Taqqu, M., Park, C., Marron, J.S.: Strengths and limitations of the wavelet spectrum method in the analysis of internet traffic, Technical Report 2004-8. Statistical and Applied Mathematical Sciences Institute, Research Triangle Park (2004)
[66] Stoev, S; Taqqu, MS; Park, C; Michailidis, G; Marron, JS, LASS: a tool for the local analysis of self-similarity, Comput. Stat. Data Anal., 50, 2447-2471, (2006) · Zbl 1445.62240
[67] Sweldens, W.: The lifting scheme: A new philosophy in biorthogonal wavelet construction, In: Laine, A., Unser, M. (eds.) Proceedings of SPIE 2569, Wavelet Applications in Signal and Image Processing III, pp. 68-79 (1995)
[68] Taqqu, MS; Teverovsky, V; Willinger, W, Estimators for long-range dependence: an empirical study, Fractals, 3, 785-798, (1995) · Zbl 0864.62061
[69] Teverovsky, V; Taqqu, M, Testing for long-range dependence in the presence of shifting means or a slowly declining trend, using a variance-type estimator, J. Time Ser. Anal., 18, 279-304, (1997) · Zbl 0934.62094
[70] Thomas, E.R., Dennis, P. F., Bracegirdle, T. J., Franzke, C.: Ice core evidence for significant 100-year regional warming on the Antarctic Peninsula, Geophys. Res. Lett. 36, L20704 (2009). doi:10.1029/2009GL040104 · Zbl 0179.47801
[71] Tomsett, AC; Toumi, R, Annual persistence in observed and modelled UK precipitation, Geophys. Res. Lett., 28, 3891-3894, (2001)
[72] Toumi, R; Syroka, J; Barnes, C; Lewis, P, Robust non-Gaussian statistics and long-range correlation of total ozone, Atmos. Sci. Lett., 2, 94-103, (2001)
[73] Trappe, W., Liu, K.: Denoising via adaptive lifting schemes, In: Aldroubi, A., Laine, M.A., Unser, M.A. (eds.) Proceedings of SPIE, Wavelet Applications in Signal and Image Processing VIII, A, vol. 4119, pp. 302-312. (2000)
[74] Tsonis, AA; Roebber, PJ; Elsner, JB, Long-range correlations in the extratropical atmospheric circulation: origins and implications, J. Clim., 12, 1534-1541, (1999)
[75] Turcotte, D.L.: Fractals and Chaos in Geology and Geophysics. Cambridge University Press, New York (1997) · Zbl 0785.58005
[76] Varotsos, C; Kirk-Davidoff, D, Long-memory processes in ozone and temperature variations at the region 60 s-60 n, Atmos. Chem. Phys., 6, 4093-4100, (2006)
[77] Veitch, D; Abry, P, A wavelet-based joint estimator of the parameters of long-range dependence, IEEE Trans. Inf. Theory, 45, 878-897, (1999) · Zbl 0945.94006
[78] Ventosa-Santaulària, D; Heres, DR; Martínez-Hernández, LC, Long-memory and the sea level-temperature relationship: a fractional cointegration approach, PloS One, 9, e113439, (2014)
[79] Vergassola, M; Frisch, U, Wavelet transforms of self-similar processes, Physica D, 54, 58-64, (1991) · Zbl 0735.76039
[80] Vidakovic, B.: Statistical Modelling by Wavelets. Wiley, New York (1999) · Zbl 0924.62032
[81] Vidakovic, BD; Katul, GG; Albertson, JD, Multiscale denoising of self-similar processes, J. Geophys. Res., 105, 27049-27058, (2000)
[82] Vyushin, DI; Fioletov, VE; Shepherd, TG, Impact of long-range correlations on trend detection in total ozone, J. Geophys. Res., 112, d14307, (2007)
[83] Whitcher, B., Jensen, M.J.: Wavelet estimation of a local long memory parameter. Explor. Geophys. 31, 94-103 (2000)
[84] Willinger, W; Taqqu, MS; Sherman, R; Wilson, DV, Self-similarity through high-variability: statistical analysis of Ethernet LAN traffic at the source level, IEEE Trans. Netw., 5, 71-86, (1997)
[85] Windsor, H.L., Toumi, R.: Scaling and persistence of UK pollution. Atmos. Environ. 35, 4545-4556 (2001) · Zbl 1445.62240
[86] Witt, A; Schumann, AY, Holocene climate variability on millennial scales recorded in greenland ice cores, Nonlinear Process. Geophys., 12, 345-352, (2005)
[87] Wolff, EW, Understanding the past-climate history from antarctica, Antarct. Sci., 17, 487-495, (2005)
[88] Wuertz, D. et al.: fARMA: ARMA Time Series Modelling, r package version 3010.79 (2013) · Zbl 1187.76481
[89] Zhang, Q., Harman, C.J., Ball, W.P.: Evaluation of methods for estimating long-range dependence (LRD) in water quality time series with missing data and irregular sampling, In: Proceedings of the American Geophysical Union Fall Meeting 2014, San Francisco (2014) · Zbl 0703.62091
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.